https://file.aiquickdraw.com/mathbot/file/1725420830782jznjg0tj.jpg

The image contains a question with two inequalities, asking to determine their solution sets:

1. **Inequality (a):**  
   \[ 4(x - 3) - 3x > 4(5 - x) \]

   We will solve this inequality step-by-step:

   \[
   4x - 12 - 3x > 20 - 4x 
   \]
   Combine like terms:
   \[
   x - 12 > 20 - 4x
   \]
   Add \(4x\) to both sides:
   \[
   x + 4x - 12 > 20 
   \]
   Combine like terms:
   \[
   5x - 12 > 20
   \]
   Add 12 to both sides:
   \[
   5x > 32
   \]
   Divide by 5:
   \[
   x > \frac{32}{5}
   \]
   So, the solution to inequality (a) is **\( x > \frac{32}{5} \)**. This means **all numbers greater than \(\frac{32}{5}\) are solutions**.

2. **Inequality (b):**  
   \[ 2(6 - x) \geq 2x + 12 \]

   Distribute on the left side:
   \[
   12 - 2x \geq 2x + 12
   \]
   Add \(2x\) to both sides:
   \[
   12 \geq 4x + 12
   \]
   Subtract 12 from both sides:
   \[
   0 \geq 4x
   \]
   Divide by 4:
   \[
   0 \geq x \quad \text{or} \quad x \leq 0
   \]
   So, the solution to inequality (b) is **\( x \leq 0 \)**. This means **all numbers less than or equal to 0 are solutions**.

### Answers:
- For (a): **\( x > \frac{32}{5} \)**
- For (b): **\( x \leq 0 \)**

Would you like more details on the solution steps? Here are some questions you might want to consider:

1. How do we handle inequalities involving both sides with variables?
2. What happens if we divide or multiply both sides of an inequality by a negative number?
3. Can you always combine like terms in inequalities in the same way as equations?
4. What is the importance of checking the solution in inequalities?
5. How would the solutions change if the inequalities had a different sign (e.g., \(\leq\) instead of \(>\))?

**Tip:** Always double-check the direction of your inequality sign when multiplying or dividing by a negative number!

For each inequality, choose the statement that describes its solution. If applicable, give the solution.

This page provides detailed solutions for solving linear inequalities, including how to manipulate and solve inequalities with variables on both sides. Perfect for students in Grades 9-11.

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